Sharpened localization of the trailing point of the Pareto record frontier

Abstract

For $d\ge2$ and iid $d$-dimensional observations $X^{(1)},X^{(2)},\dots$ with independent Exponential$(1)$ coordinates, we revisit the study by Fill and Naiman (Electron. J. Probab., 2020) of the boundary (relative to the closed positive orthant), or "frontier", $F_n$ of the closed Pareto record-setting (RS) region \[ \mbox{RS}_n:=\{0\le x\in{\mathbb R}^d:x\not\prec X^{(i)}\mbox{\ for all $1\le i\le n$}\} \] at time $n$, where $0\le x$ means that $0\le x_j$ for $1\le j\le d$ and $x\prec y$ means that $x_j<y_j$ for $1\le j\le d$. With $x_+:=\sum_{j=1}^d x_j$, let \[ F_n^-:=\min\{x_+:x\in F_n\}\quad\mbox{and}\quad F_n^+:=\max\{x_+:x\in F_n\}. \] Almost surely, there are for each $n$ unique vectors $\lambda_n\in F_n$ and $\tau_n\in F_n$ such that $F_n^+=(\lambda_n)_+$ and $F_n^-=(\tau_n)_+$; we refer to $\lambda_n$ and $\tau_n$ as the leading and trailing points, respectively, of the frontier. Fill and Naiman provided rather sharp information about the typical and almost sure behavior of $F^+$, but somewhat crude information about $F^-$, namely, that for any $\varepsilon >0$ and $c_n\to\infty$ we have \[ {\mathbb P}(F_n^- -\ln n\in (-(2+\varepsilon)\ln\ln\ln n,c_n))\to 1 \] (describing typical behavior) and almost surely \[ \limsup \frac{F_n^- - \ln n}{\ln \ln n} \le 0 \quad \mbox{and} \quad \liminf \frac{F_n^- - \ln n}{\ln \ln \ln n} \in [-2, -1]. \] In this paper we use the theory of generators (minima of $F_n$) together with the first- and second-moment methods to improve considerably the trailing-point location results to \[ F_n^- - (\ln n - \ln \ln \ln n) \overset{\mathrm{P}}{\longrightarrow} - \ln(d - 1) \] (describing typical behavior) and, for $d \ge 3$, almost surely \begin{align*} &\limsup [F_n^- - (\ln n - \ln \ln \ln n)] \leq -\ln(d - 2) + \ln 2 \\ \mbox{and }&\liminf [F_n^- - (\ln n - \ln \ln \ln n)] \ge - \ln d - \ln 2. \end{align*}